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Nov
14
answered Ratio of geodesic segments on the sphere
Nov
14
comment Best-fitting plane
@J.M.: Ah, I see, sorry -- I thought you were suggesting to apply SVD instead of eigendecomposition to the covariance matrix :-)
Nov
14
comment What happens to a regular Markov matrix that has more than one steady state/stationary distribution?
The regularity ensures that the process is irreducible and aperiodic, and in that case there's a single stationary distribution.
Nov
14
comment Equivalence relation on the functions from $\mathbb{N}$ to $\mathbb{R}_{\geq 0}$
Yes, but that's what I wrote: a counterexample to the inequality means the inequality doesn't hold.
Nov
14
comment Best-fitting plane
@J.M.: I don't understand -- since $M$ is symmetric, isn't that the same thing?
Nov
14
comment Best-fitting plane
Yes, this is the correct approach for the best-fitting plane. To get useful responses where to find a suitable eigenvalue solver, you should probably tell us more about your computing environment and requirements.
Nov
14
answered Equivalence relation on the functions from $\mathbb{N}$ to $\mathbb{R}_{\geq 0}$
Nov
14
revised Slowing down the divergence
deleted 5 characters in body
Nov
14
comment Equivalence relation on the functions from $\mathbb{N}$ to $\mathbb{R}_{\geq 0}$
@Peter: Sorry, yes, the $g$ was supposed to be an $f$ -- I'll try to write down the counterexample if I can still remember it :-)
Nov
14
revised Slowing down the divergence
deleted 3 characters in body
Nov
14
comment Slowing down the divergence
Why is $b$ nonincreasing?
Nov
14
answered Slowing down the divergence
Nov
14
answered A unitary Matrix with positively definite matrices, truly behaving as unit under multiplicity?
Nov
14
comment Equivalence relation on the functions from $\mathbb{N}$ to $\mathbb{R}_{\geq 0}$
Why do you say "is also a monotone function"? What else is a monotone function? Did you forget to specify that $f$ and $g$ are monotone? If there are no further conditions on $f$ and $g$, the statement is false, since you can construct an $f$ for which there is a counterexample to $f(n)\le Cg(Cn+C)+Cn+C$ for each $C\in\mathbb N$ (and presumably $C\in\mathbb N$ since otherwise $g(Cn+C)$ isn't defined).
Nov
14
answered Conway's napkin puzzle
Nov
13
comment Distribution of sum of two random variables
@cardinal: Ha -- I was myself implicitly thinking of independent normally distributed variables :-) Yes, the quick edit thing almost worked -- I was in time to open the comment for editing, but not to save the change :-)
Nov
13
comment Distribution of sum of two random variables
You seem to be assuming that the mean and variance determine the distribution, or perhaps you're implicitly thinking of normally distributed variables. All you can determine from the mean and variance of the variables is the mean and variance of the sum.
Nov
13
comment Are these three ways of normalizing a set/vector of numbers 'equivalent'?
@Nupul: I see. In that case, what I wrote doesn't hold literally for #2, since it's not linear -- but the general conclusion that the methods aren't equivalent is still valid.
Nov
13
comment Are these three ways of normalizing a set/vector of numbers 'equivalent'?
@Nupul: A transformation is linear if it's described by a linear polynomial, that is, $s'=\alpha s+\beta$ as in my answer. You can write each of your three transformations like this; for instance, for method $3$ (with rating $1$), $\alpha=-1/(\mathrm{worst}-\mathrm{best})$ and $\beta=\mathrm{worst}/(\mathrm{worst}-\mathrm{best})$. You can't write the method "Another way" like this, because $s$ is in the denominator. I'm not sure what you mean by this method being done with method #2; I'm basing this on the equation you wrote for this method, $s'_i=\min(S)/s_i$, with $s_i$ in the denominator.
Nov
13
revised Find control point on piecewise quadratic Bézier curve
corrected spelling